Tips to remember the weighted Harmonic mean?

CFA Level II Quant
#1

Can’t seem to memorize this one!

Thanks!

#2

The (usual, unweighted) harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. If memory serve, the weighted harmonic mean is the reciprocal of the weighted average of the reciprocals. Perhaps that’ll help a bit.

#3

Not sure if this will be helpful, but I always think “inverse of the inverses”.

Given a simple 2-sample case of x1 & x2 with w1 & w2:

  1. Take the inverse of x1, weight it by w1. Take the inverse of x2, weight it by w2 (or 1-w1 in this case).
  2. Sum the results.
  3. Then take the inverse of the result.
  4. Multiply by the number of x terms (in this case 2).

You can swap steps 3 & 4 by DIVIDING the results by N, THEN taking the inverse. Same difference.

I realize this doesn’t necessarily help you memorize the formula, but if you just remember “inverse of the (summed weighted) inverses”, you will be almost 100% there (just don’t forget to multiply the result by N).

General formula: N / summation(wi/xi)

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